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Taylor series can create harmonics of the frequencies in an input signal. I'm wondering if it is likewise possible to use Volterra series to create sub-harmonics of the frequencies in that signal.

Some degree of finesse is required in posing the question:

Taylor series don't only create harmonics; there's intermodulation distortion as well. Likewise, it's ok with me if a suitable Volterra series doesn't only create sub-harmonics, but also creates something similar to intermodulation distortion, or even some degree of harmonic distortion.

However, I would like to exclude trivial cases where the input signal gets so wrecked that it ends up looking like white noise, which would obviously create sub-harmonics (along with every other frequency you can imagine!). I'm not sure how to formalize this requirement precisely, other than to say that I hope it's clear what I'm driving at.

Here's one way to formalize the behaviour I want:

  1. Given an input $\cos(\omega t)$, yields an output containing $\cos(\frac{\omega}{n} t)$ for some fixed $n$, ideally a natural number
  2. Given a general input, a sub-harmonic is generated corresponding to every frequency in the input
  3. Other "intermodulation distortion"-type artifacts are allowed, "within reason"

The answer can be expressed for either discrete or continuous signals.

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  • $\begingroup$ interesting question. i don't know for sure, but i suspect that you cannot generate sub-harmonics without a time-varying component in the algorithm. Volterra is non-linear, but is also time-invariant. $\endgroup$ – robert bristow-johnson Nov 14 '15 at 14:25
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Check out this paper. I would have made a comment but not high enough rep.

http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=1211087&url=http%3A%2F%2Fieeexplore.ieee.org%2Fiel5%2F81%2F27258%2F01211087

Looks like you need multiple in to get subharmonics in Volterra series

The abstract states "Subharmonic generation is a complex nonlinear phenomenon which can arise from nonlinear oscillations, bifurcation and chaos. It is well known that single-input-single-output Volterra series cannot currently be applied to model systems which exhibit subharmonics. A new modeling alternative is introduced in this paper which overcomes these restrictions by using local multiple input single output Volterra models. The generalized frequency-response functions can then be applied to interpret systems with subharmonics in the frequency domain."

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  • $\begingroup$ Thanks, although I'm a bit confused. When we talk about a MISO system, which input are we generating subharmonics of? Both? I can't see the paper, so I'm a bit confused here. $\endgroup$ – Mike Battaglia Nov 19 '15 at 9:26
  • $\begingroup$ full disclosure - I have never done MISO Volterra as I can limit my analysis that I do to SISO but I would imagine that you could look at analygous to a subharmonic mixer if you have ever came across one. In this case, a low frequency tone is mixed with a high frequency one, say something like LO = 10 GHz and RF = 100.1 GHz, the mixer product, IF would be at multiple subharmonics but if you wanted the 10 harmonic of the LO, IF would be 100 MHz ( this is done a lot with millimeter wave systems) - but really, I would read the paper I linked here. $\endgroup$ – johnnymopo Nov 20 '15 at 3:05

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